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72-457: It consists of libmad, a software library, and madplay, a command-line program for MP3 playback. libmad is notable for using only fixed-point arithmetic while madplay is notable for its ReplayGain support. Development ended decades ago, and vulnerabilities identified since then have gone unpatched. This free and open-source software article is a stub . You can help Misplaced Pages by expanding it . This multimedia software -related article

144-405: A /10 , such as 1/100 or 37/1000, can be exactly represented in fixed point with a power-of-ten scaling factor 1/10 with any n ≥ m . This decimal format can also represent any binary fraction a /2 , such as 1/8 (0.125) or 17/32 (0.53125). More generally, a rational number a / b , with a and b relatively prime and b positive, can be exactly represented in binary fixed point only if b

216-549: A fixed-point representation can have error less than 1.2 × 10 , whereas the standard floating-point representation may have error up to 596 × 10 — because 9 of the bits are wasted with the sign and exponent of the dynamic scaling factor. Specifically, comparing 32-bit fixed-point to floating-point audio, a recording requiring less than 40 dB of headroom has a higher signal-to-noise ratio using 32-bit fixed. Programs using fixed-point computations are usually more portable than those using floating-point, since they do not depend on

288-504: A four-bit opcode can specify up to sixteen different ALU operations. Generally, an ALU opcode is not the same as a machine language instruction , though in some cases it may be directly encoded as a bit field within such instructions. The status outputs are various individual signals that convey supplemental information about the result of the current ALU operation. General-purpose ALUs commonly have status signals such as: The status inputs allow additional information to be made available to

360-462: A fractional amount of hours as an integer multiple of ten-minute intervals. Fixed-point number representation is often contrasted to the more complicated and computationally demanding floating-point representation . In the fixed-point representation, the fraction is often expressed in the same number base as the integer part, but using negative powers of the base b . The most common variants are decimal (base 10) and binary (base 2). The latter

432-407: A fractional amount of hours may be represented as an integer number of seconds; that is, as a fixed-point number with scale factor of 1/3600. Even with the most careful rounding, fixed-point values represented with a scaling factor S may have an error of up to ±0.5 in the stored integer, that is, ±0.5 S in the value. Therefore, smaller scaling factors generally produce more accurate results. On

504-496: A multiply instruction that includes renormalization—the scaling conversion of the product from 2 n −2 to n −1 fraction bits. If the CPU does not provide that feature, the programmer must save the product in a large enough register or temporary variable, and code the renormalization explicitly. Overflow happens when the result of an arithmetic operation is too large to be stored in the designated destination area. In addition and subtraction,

576-428: A signed integer in two's complement representation with an implicit scaling factor as above. The sign of the value will always be indicated by the first stored bit (1 = negative, 0 = non-negative), even if the number of fraction bits is greater than or equal to the total number of bits. For example, the 8-bit signed binary integer (11110101) 2 = −11, taken with -3, +5, and +12 implied fraction bits, would represent

648-407: A variable or formula may not appear explicitly in the program. Good programming practice then requires that it be provided in the documentation , at least as a comment in the source code . For greater efficiency, scaling factors are often chosen to be powers (positive or negative) of the base b used to represent the integers internally. However, often the best scaling factor is dictated by

720-571: A very simple 8-bit ALU: Mathematician John von Neumann proposed the ALU concept in 1945 in a report on the foundations for a new computer called the EDVAC . The cost, size, and power consumption of electronic circuitry was relatively high throughout the infancy of the Information Age . Consequently, all early computers had a serial ALU that operated on one data bit at a time although they often presented

792-596: A wider word size to programmers. The first computer to have multiple parallel discrete single-bit ALU circuits was the 1951 Whirlwind I , which employed sixteen such "math units" to enable it to operate on 16-bit words. In 1967, Fairchild introduced the first ALU-like device implemented as an integrated circuit, the Fairchild 3800, consisting of an eight-bit arithmetic unit with accumulator. It only supported adds and subtracts but no logic functions. Full integrated-circuit ALUs soon emerged, including four-bit ALUs such as

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864-423: Is 22×32/100 = 7.04. If the result is not exact, the error introduced by the rounding can be reduced or even eliminated by converting the dividend to a smaller scaling factor. For example, if r = 1.23 is represented as 123 with scaling 1/100, and s = 6.25 is represented as 6250 with scaling 1/1000, then simple division of the integers yields 123÷6250 = 0 (rounded) with scaling factor (1/100)/(1/1000) = 10. If r

936-496: Is a stub . You can help Misplaced Pages by expanding it . Fixed-point arithmetic In computing , fixed-point is a method of representing fractional (non-integer) numbers by storing a fixed number of digits of their fractional part. Dollar amounts, for example, are often stored with exactly two fractional digits, representing the cents (1/100 of dollar). More generally, the term may refer to representing fractional values as integer multiples of some fixed small unit, e.g.

1008-488: Is a fixed-point environment having each array (block) of fixed-point data be scaled with a common exponent in a single word. A common use of decimal fixed-point is for storing monetary values, for which the complicated rounding rules of floating-point numbers are often a liability. For example, the open-source money management application GnuCash , written in C, switched from floating-point to fixed-point as of version 1.6, for this reason. Binary fixed-point (binary scaling)

1080-421: Is a power of 2; and in decimal fixed point only if b has no prime factors other than 2 and/or 5. Fixed-point computations can be faster and/or use less hardware than floating-point ones. If the range of the values to be represented is known in advance and is sufficiently limited, fixed point can make better use of the available bits. For example, if 32 bits are available to represent a number between 0 and 1,

1152-436: Is an algorithm that operates on integers which are larger than the ALU word size. To do this, the algorithm treats each integer as an ordered collection of ALU-size fragments, arranged from most-significant (MS) to least-significant (LS) or vice versa. For example, in the case of an 8-bit ALU, the 24-bit integer 0x123456 would be treated as a collection of three 8-bit fragments: 0x12 (MS), 0x34 , and 0x56 (LS). Since

1224-424: Is commonly known also as binary scaling . Thus, if n fraction digits are stored, the value will always be an integer multiple of b . Fixed-point representation can also be used to omit the low-order digits of integer values, e.g. when representing large dollar values as multiples of $ 1000. When decimal fixed-point numbers are displayed for human reading, the fraction digits are usually separated from those of

1296-437: Is first converted to 1,230,000 with scaling factor 1/1000000, the result will be 1,230,000÷6250 = 197 (rounded) with scale factor 1/1000 (0.197). The exact value 1.23/6.25 is 0.1968. In fixed-point computing it is often necessary to convert a value to a different scaling factor. This operation is necessary, for example: To convert a number from a fixed point type with scaling factor R to another type with scaling factor S ,

1368-407: Is generally easier and safer to select scaling factors and word sizes so as to exclude the possibility of overflow, or to check the operands for excessive values before executing the operation. Arithmetic logic unit In computing , an arithmetic logic unit ( ALU ) is a combinational digital circuit that performs arithmetic and bitwise operations on integer binary numbers . This

1440-415: Is in contrast to a floating-point unit (FPU), which operates on floating point numbers. It is a fundamental building block of many types of computing circuits, including the central processing unit (CPU) of computers, FPUs, and graphics processing units (GPUs). The inputs to an ALU are the data to be operated on, called operands , and a code indicating the operation to be performed; the ALU's output

1512-406: Is more natural for the problem. Examples of the latter are accounting of dollar amounts, when fractions of cents must be rounded to whole cents in strictly prescribed ways; and the evaluation of functions by table lookup , or any application where rational numbers need to be represented without rounding errors (which fixed-point does but floating-point cannot). Fixed-point representation is still

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1584-460: Is operating, external circuits apply signals to the ALU inputs and, in response, the ALU produces and conveys signals to external circuitry via its outputs. A basic ALU has three parallel data buses consisting of two input operands ( A and B ) and a result output ( Y ). Each data bus is a group of signals that conveys one binary integer number. Typically, the A, B and Y bus widths (the number of signals comprising each bus) are identical and match

1656-408: Is possible by adding a 'rounding addend' of half of the scaling factor before shifting; The proof: round(x/y) = floor(x/y + 0.5) = floor((x + y/2)/y) = shift-of-n(x + 2^(n-1)) A similar method is usable in any scaling. To divide two fixed-point numbers, one takes the integer quotient of their underlying integers, and assumes that the scaling factor is the quotient of their scaling factors. In general,

1728-410: Is referred to as the "status register" or "condition code register". Depending on the ALU operation being performed, some status register bits may be changed and others may be left unmodified. For example, in bitwise logical operations such as AND and OR, the carry status bit is typically not modified as it is not relevant to such operations. In CPUs, the stored carry-out signal is usually connected to

1800-407: Is repeated for all operand fragments so as to generate a complete collection of partials, which is the result of the multiple-precision operation. In arithmetic operations (e.g., addition, subtraction), the algorithm starts by invoking an ALU operation on the operands' LS fragments, thereby producing both a LS partial and a carry out bit. The algorithm writes the partial to designated storage, whereas

1872-425: Is the result of the performed operation. In many designs, the ALU also has status inputs or outputs, or both, which convey information about a previous operation or the current operation, respectively, between the ALU and external status registers . An ALU has a variety of input and output nets , which are the electrical conductors used to convey digital signals between the ALU and external circuitry. When an ALU

1944-462: The Am2901 and 74181 . These devices were typically " bit slice " capable, meaning they had "carry look ahead" signals that facilitated the use of multiple interconnected ALU chips to create an ALU with a wider word size. These devices quickly became popular and were widely used in bit-slice minicomputers. Microprocessors began to appear in the early 1970s. Even though transistors had become smaller, there

2016-546: The discrete cosine transform algorithms used to compress JPEG images. Electronic instruments such as electricity meters and digital clocks often use polynomials to compensate for introduced errors, e.g. from temperature or power supply voltage. The coefficients are produced by polynomial regression . Binary fixed-point polynomials can utilize more bits of precision than floating-point, and do so in fast code using inexpensive CPUs. Accuracy, crucial for instruments, compares well to equivalent-bit floating-point calculations, if

2088-430: The ALU inputs. Typically, the external circuitry employs sequential logic to generate the signals that control ALU operation. The external sequential logic is paced by a clock signal of sufficiently low frequency to ensure enough time for the ALU outputs to settle under worst-case conditions (i.e., conditions resulting in the maximum possible propagation delay). For example, a CPU starts an addition operation by routing

2160-414: The ALU when performing an operation. Typically, this is a single "carry-in" bit that is the stored carry-out from a previous ALU operation. An ALU is a combinational logic circuit, meaning that its outputs will change asynchronously in response to input changes. In normal operation, stable signals are applied to all of the ALU inputs and, when enough time (known as the " propagation delay ") has passed for

2232-400: The ALU's carry-in net. This facilitates efficient propagation of carries (which may represent addition carries, subtraction borrows, or shift overflows) when performing multiple-precision operations, as it eliminates the need for software-management of carry propagation (via conditional branching, based on the carry status bit). In integer arithmetic computations, multiple-precision arithmetic

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2304-512: The LS bit of each partial—which is conveyed via the stored carry bit—must be obtained from the MS bit of the previously left-shifted, less-significant operand. Conversely, operands are processed MS first in right-shift operations because the MS bit of each partial must be obtained from the LS bit of the previously right-shifted, more-significant operand. In bitwise logical operations (e.g., logical AND, logical OR),

2376-449: The accuracy δ = S /2 of values that could be represented in 16-bit signed binary fixed point format, depending on the number f of implied fraction bits. Fixed-point formats with scaling factors of the form 2 -1 (namely 1, 3, 7, 15, 31, etc.) have been said to be appropriate for image processing and other digital signal processing tasks. They are supposed to provide more consistent conversions between fixed- and floating-point values than

2448-488: The application. Thus one often uses scaling factors that are powers of 10 (e.g. 1/100 for dollar values), for human convenience, even when the integers are represented internally in binary. Decimal scaling factors also mesh well with the metric (SI) system , since the choice of the fixed-point scaling factor is often equivalent to the choice of a unit of measure (like centimeters or microns instead of meters ). However, other scaling factors may be used occasionally, e.g.

2520-628: The availability of an FPU. This advantage was particularly strong before the IEEE Floating Point Standard was widely adopted, when floating-point computations with the same data would yield different results depending on the manufacturer, and often on the computer model. Many embedded processors lack an FPU, because integer arithmetic units require substantially fewer logic gates and consume much smaller chip area than an FPU; and software emulation of floating-point on low-speed devices would be too slow for most applications. CPU chips for

2592-427: The base b ; whereas their absolute resolution varies by many orders of magnitude, like the values themselves. In many cases, the rounding and truncation errors of fixed-point computations are easier to analyze than those of the equivalent floating-point computations. Applying linearization techniques to truncation, such as dithering and/or noise shaping is more straight-forward within fixed-point arithmetic. On

2664-455: The earlier personal computers and game consoles , like the Intel 386 and 486SX , also lacked an FPU. The absolute resolution (difference between successive values) of any fixed-point format is constant over the whole range, namely the scaling factor S . In contrast, the relative resolution of a floating-point format is approximately constant over their whole range, varying within a factor of

2736-477: The first division requires rounding and therefore the result is not exact. For example, division of 3456 scaled by 1/100 (34.56) and 1234 scaled by 1/1000 (1.234) yields the integer 3456÷1234 = 3 (rounded) with scale factor (1/100)/(1/1000) = 10, that is, 30. As another example, the division of the first number by 155 implicitly scaled by 1/32 (155/32 = 4.84375) yields the integer 3456÷155 = 22 (rounded) with implicit scaling factor (1/100)/(1/32) = 32/100 = 8/25, that

2808-407: The first number by 155 implicitly scaled by 1/32 (155/32 = 4.84375) yields the integer 123×155 = 19065 with implicit scaling factor (1/1000)×(1/32) = 1/32000, that is 19065/32000 = 0.59578125. In binary, it is common to use a scaling factor that is a power of two. After the multiplication, the scaling factor can be divided away by shifting right. Shifting is simple and fast in most computers. Rounding

2880-444: The fixed-point polynomials are evaluated using Horner's method (e.g. y = (( ax + b ) x + c ) x + d ) to reduce the number of times that rounding occurs, and the fixed-point multiplications utilize rounding addends. To add or subtract two values with the same implicit scaling factor, it is sufficient to add or subtract the underlying integers; the result will have their common implicit scaling factor, can thus can be stored in

2952-406: The integer 3075 then must be multiplied by 1/100, that is, divided by 100, to yield either 31 (0.31) or 30 (0.30), depending on the rounding policy used. Similarly, the operation r ← r / s will require dividing the integers and explicitly multiplying the quotient by S . Rounding and/or overflow may occur here too. To convert a number from floating point to fixed point, one may multiply it by

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3024-806: The integer part by a radix character (usually '.' in English, but ',' or some other symbol in many other languages). Internally, however, there is no separation, and the distinction between the two groups of digits is defined only by the programs that handle such numbers. Fixed-point representation was the norm in mechanical calculators . Since most modern processors have fast floating-point unit (FPU), fixed-point representations in processor-based implementations are now used only in special situations, such as in low-cost embedded microprocessors and microcontrollers ; in applications that demand high speed or low power consumption or small chip area, like image , video , and digital signal processing ; or when their use

3096-461: The integer, or appending zero digits. However, this operation must preserve the sign of the number. In two's complement representation, that means extending the sign bit as in arithmetic shift operations. If S does not divide R (in particular, if the new scaling factor S is greater than the original R ), the new integer may have to be rounded . In particular, if r and s are fixed-point variables with implicit scaling factors R and S ,

3168-464: The last 3 decimal digits are implicitly assumed to be a decimal fraction), and the value 1 230 000 can be represented as 1230 with an implicit scaling factor of 1000 (with "minus 3" implied decimal fraction digits, that is, with 3 implicit zero digits at right). This representation allows standard integer arithmetic logic units to perform rational number calculations. Negative values are usually represented in binary fixed-point format as

3240-427: The native word size of the external circuitry (e.g., the encapsulating CPU or other processor). The opcode input is a parallel bus that conveys to the ALU an operation selection code, which is an enumerated value that specifies the desired arithmetic or logic operation to be performed by the ALU. The opcode size (its bus width) determines the maximum number of distinct operations the ALU can perform; for example,

3312-567: The next clock, are allowed to propagate through the ALU and to the destination register while the CPU waits for the next clock. When the next clock arrives, the destination register stores the ALU result and, since the ALU operation has completed, the ALU inputs may be set up for the next ALU operation. A number of basic arithmetic and bitwise logic functions are commonly supported by ALUs. Basic, general purpose ALUs typically include these operations in their repertoires: ALU shift operations cause operand A (or B) to shift left or right (depending on

3384-451: The norm for field-programmable gate array (FPGA) implementations, as floating-point support in an FPGA requires significantly more resources than fixed-point support. A fixed-point representation of a fractional number is essentially an integer that is to be implicitly multiplied by a fixed scaling factor. For example, the value 1.23 can be stored in a variable as the integer value 1230 with implicit scaling factor of 1/1000 (meaning that

3456-406: The opcode) and the shifted operand appears at Y. Simple ALUs typically can shift the operand by only one bit position, whereas more complex ALUs employ barrel shifters that allow them to shift the operand by an arbitrary number of bits in one operation. In all single-bit shift operations, the bit shifted out of the operand appears on carry-out; the value of the bit shifted into the operand depends on

3528-588: The operand fragments may be processed in any arbitrary order because each partial depends only on the corresponding operand fragments (the stored carry bit from the previous ALU operation is ignored). Although it is possible to design ALUs that can perform complex functions, this is usually impractical due to the resulting increases in circuit complexity, power consumption, propagation delay, cost and size. Consequently, ALUs are typically limited to simple functions that can be executed at very high speeds (i.e., very short propagation delays), with more complex functions being

3600-422: The operands from their sources (typically processor registers ) to the ALU's operand inputs, while simultaneously applying a value to the ALU's opcode input that configures it to perform an addition operation. At the same time, the CPU enables the destination register to store the ALU output (the resulting sum from the addition operation) upon operation completion. The ALU's input signals, which are held stable until

3672-403: The operation r ← r × s require multiplying the respective integers and explicitly dividing the result by S . The result may have to be rounded, and overflow may occur. For example, if the common scaling factor is 1/100, multiplying 1.23 by 0.25 entails multiplying 123 by 25 to yield 3075 with an intermediate scaling factor of 1/10000. In order to return to the original scaling factor 1/100,

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3744-462: The other hand, a smaller scaling factor means a smaller range of the values that can be stored in a given program variable. The maximum fixed-point value that can be stored into a variable is the largest integer value that can be stored into it, multiplied by the scaling factor; and similarly for the minimum value. For example, the table below gives the implied scaling factor S , the minimum and maximum representable values V min and V max , and

3816-432: The other hand, the use of fixed point requires greater care by the programmer. Avoidance of overflow requires much tighter estimates for the ranges of variables and all intermediate values in the computation, and often also extra code to adjust their scaling factors. Fixed-point programming normally requires the use of integer types of different widths . Fixed-point applications can make use of block floating point , which

3888-410: The partial is written to designated storage. This process repeats until all operand fragments have been processed, resulting in a complete collection of partials in storage, which comprise the multi-precision arithmetic result. In multiple-precision shift operations, the order of operand fragment processing depends on the shift direction. In left-shift operations, fragments are processed LS first because

3960-415: The processor's state machine typically stores the carry out bit to an ALU status register. The algorithm then advances to the next fragment of each operand's collection and invokes an ALU operation on these fragments along with the stored carry bit from the previous ALU operation, thus producing another (more significant) partial and a carry out bit. As before, the carry bit is stored to the status register and

4032-451: The responsibility of external circuitry. For example: An ALU is usually implemented either as a stand-alone integrated circuit (IC), such as the 74181 , or as part of a more complex IC. In the latter case, an ALU is typically instantiated by synthesizing it from a description written in VHDL , Verilog or some other hardware description language . For example, the following VHDL code describes

4104-401: The result may require one bit more than the operands. In multiplication of two unsigned integers with m and n bits, the result may have m + n bits. In case of overflow, the high-order bits are usually lost, as the un-scaled integer gets reduced modulo 2 where n is the size of the storage area. The sign bit, in particular, is lost, which may radically change the sign and the magnitude of

4176-411: The same program variables as the operands. These operations yield the exact mathematical result, as long as no overflow occurs—that is, as long as the resulting integer can be stored in the receiving program variable . If the operands have different scaling factors, then they must be converted to a common scaling factor before the operation. To multiply two fixed-point numbers, it suffices to multiply

4248-409: The scaling factor S , then round the result to the nearest integer. Care must be taken to ensure that the result fits in the destination variable or register. Depending on the scaling factor and storage size, and on the range input numbers, the conversion may not entail any rounding. To convert a fixed-point number to floating-point, one may convert the integer to floating-point and then divide it by

4320-657: The scaling factor S . This conversion may entail rounding if the integer's absolute value is greater than 2 (for binary single-precision IEEE floating point) or of 2 (for double-precision). Overflow or underflow may occur if | S | is very large or very small, respectively. Typical processors do not have specific support for fixed-point arithmetic. However, most computers with binary arithmetic have fast bit shift instructions that can multiply or divide an integer by any power of 2; in particular, an arithmetic shift instruction. These instructions can be used to quickly change scaling factors that are powers of 2, while preserving

4392-737: The sign of the number. Early computers like the IBM 1620 and the Burroughs B3500 used a binary-coded decimal (BCD) representation for integers, namely base 10 where each decimal digit was independently encoded with 4 bits. Some processors, such as microcontrollers, may still use it. In such machines, conversion of decimal scaling factors can be performed by bit shifts and/or by memory address manipulation. Some DSP architectures offer native support for specific fixed-point formats, for example, signed n -bit numbers with n −1 fraction bits (whose values may range between −1 and almost +1). The support may include

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4464-437: The signals to propagate through the ALU circuitry, the result of the ALU operation appears at the ALU outputs. The external circuitry connected to the ALU is responsible for ensuring the stability of ALU input signals throughout the operation, and for allowing sufficient time for the signals to propagate through the ALU circuitry before sampling the ALU outputs. In general, external circuitry controls an ALU by applying signals to

4536-432: The size of a fragment exactly matches the ALU word size, the ALU can directly operate on this "piece" of operand. The algorithm uses the ALU to directly operate on particular operand fragments and thus generate a corresponding fragment (a "partial") of the multi-precision result. Each partial, when generated, is written to an associated region of storage that has been designated for the multiple-precision result. This process

4608-450: The two underlying integers, and assume that the scaling factor of the result is the product of their scaling factors. The result will be exact, with no rounding, provided that it does not overflow the receiving variable. For example, multiplying the numbers 123 scaled by 1/1000 (0.123) and 25 scaled by 1/10 (2.5) yields the integer 123×25 = 3075 scaled by (1/1000)×(1/10) = 1/10000, that is 3075/10000 = 0.3075. As another example, multiplying

4680-404: The type of shift. Upon completion of each ALU operation, the ALU's status output signals are usually stored in external registers to make them available for future ALU operations (e.g., to implement multiple-precision arithmetic ) and for controlling conditional branching . The bit registers that store the status output signals are often collectively treated as a single, multi-bit register, which

4752-448: The underlying integer must be multiplied by the ratio R / S . Thus, for example, to convert the value 1.23 = 123/100 from scaling factor R =1/100 to one with scaling factor S =1/1000, the integer 123 must be multiplied by (1/100)/(1/1000) = 10, yielding the representation 1230/1000. If the scaling factor is a power of the base used internally to represent the integer, changing the scaling factor requires only dropping low-order digits of

4824-410: The usual 2 scaling. The Julia programming language implements both versions. Any binary fraction a /2 , such as 1/16 or 17/32, can be exactly represented in fixed-point, with a power-of-two scaling factor 1/2 with any n ≥ m . However, most decimal fractions like 0.1 or 0.123 are infinite repeating fractions in base 2. and hence cannot be represented that way. Similarly, any decimal fraction

4896-433: The value. Some processors can set a hardware overflow flag and/or generate an exception on the occurrence of an overflow. Some processors may instead provide saturation arithmetic : if the result of an addition or subtraction were to overflow, they store instead the value with the largest magnitude that can fit in the receiving area and has the correct sign. However, these features are not very useful in practice; it

4968-501: The values −11/2 = −88, −11/2 = −0. 343 75 , and −11/2 = −0. 002 685 546 875 , respectively. Alternatively, negative values can be represented by an integer in the sign-magnitude format, in which case the sign is never included in the number of implied fraction bits. This variant is more commonly used in decimal fixed-point arithmetic. Thus the signed 5-digit decimal integer (−00025) 10 , taken with -3, +5, and +12 implied decimal fraction digits, would represent

5040-449: The values −25/10 = −25000, −25/10 = −0.00025, and −25/10 = −0. 000 000 000 025 , respectively. A program will usually assume that all fixed-point values that will be stored into a given variable, or will be produced by a given instruction , will have the same scaling factor. This parameter can usually be chosen by the programmer depending on the precision needed and range of values to be stored. The scaling factor of

5112-530: Was sometimes insufficient die space for a full-word-width ALU and, as a result, some early microprocessors employed a narrow ALU that required multiple cycles per machine language instruction. Examples of this includes the popular Zilog Z80 , which performed eight-bit additions with a four-bit ALU. Over time, transistor geometries shrank further, following Moore's law , and it became feasible to build wider ALUs on microprocessors. Modern integrated circuit (IC) transistors are orders of magnitude smaller than those of

5184-463: Was widely used from the late 1960s to the 1980s for real-time computing that was mathematically intensive, such as flight simulation and in nuclear power plant control algorithms. It is still used in many DSP applications and custom-made microprocessors. Computations involving angles would use binary angular measurement . Binary fixed point is used in the STM32G4 series CORDIC co-processors and in

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